EMPIRICAL CHARACTERISTIC FUNCTION APPROACH TO GOODNESS-OF-FIT TESTS FOR THE GENERALIZED EXPONENTIAL DISTRIBUTION

Fit Tests for the Generalized Exponential Distribution

Suzanne Abdel-Rahman Allam Mathematical Statistics Department

Institute of Statistical Studies and Research Cairo University, Egypt

[email protected]

Abstract

The goodness-of-fit test of Towhidi and Salmanpour (2007), which is based on the empirical characteristic function is used for testing the fit of the generalized exponential distribution. Monte Carlo procedures are employed to obtain the empirical percentage points for the test statistic. In addition, an extensive power study is conducted to compare the performance of this test against the well known goodness-of-fit tests which are based on empirical distribution function.

Keywords: goodness-of-fit tests, empirical characteristic function, generalized exponential distribution, maximum likelihood estimation, empirical distribution function.

1. Introduction

A statistical problem encountered in many areas of research is the need to assess whether a sample of observations comes from a specified distribution. Typically such situations are known as 'Goodness of Fit' problems. Many goodness-of-fit tests were introduced in the literature. The tests based on the empirical distribution function (EDF) are the most commonly used goodness-of-fit tests. In addition, there are goodness-of-fit tests that are based on the empirical characteristic function (ECF).

The generalized exponential distribution has been introduced and studied quite extensively by Gupta and Kundu (1999, 2001, 2002, 2003). It is observed that this distribution can be considered for situations where a skewed distribution for a non-negative random variable is needed. Also, it is observed that it can be used quite effectively to analyze lifetime data in place of gamma, Weibull and log-normal distributions. The generalized exponential distribution has the following distribution function:

( ; , ) (1 x)

F x _{   }e _{; , 0 (1.1)}

for x0 and 0 otherwise. The corresponding density function is:

1

( ; , ) (1 x) x

f x _{   }e e _{; , 0 (1.2)}

for x0 and 0 otherwise. In addition, the CF of the generalized exponential distribution is given by:

 

1

( )

1

it t

it

 

  

 

 _{ }

 

 

 

  _{ }

 

Here



is the shape parameter and  is the scale parameter. If the shape parameter  1, then the generalized exponential distribution coincides with the exponential distribution with a scale parameter . The generalized exponential distribution with the two parameters



and  will be denoted by GE(



, ).

Hassan (2005) used different EDF statistics for testing the goodness of fit for the GE( , ) with unknown parameters. She obtained the critical values for these test statistics after estimating the unknown parameters by ML method. In addition, Hassan (2005) conducted a power comparison between the Kolmogorov-Smirnov (K-S), Cramer-von Mises (C-M), Anderson-Darling(A-D), Watson (WA), and Liao and Shimokawa (L-S) tests for the GE( , ) with the two parameters unknown. She considered different alternatives that are; the standard normal, Weibull, Gamma, exponential, chi-square and uniform distributions. From this power study, Hassan (2005) concluded that the L-S test is generally superior to other tests. Also, she mentioned that the WA test is not appearing to be powerful across this group of alternative distributions.

The present work is interested in applying the test of Towhidi and Salmanpour (2007) for testing the null hypothesis:

0

H : The sample x x₁, ,...,₂ xn comes from GE(



, ) with unknown parameters



and . (1.4)

The unknown parameters of this distribution will be estimated by the maximum likelihood (ML) method. Also, a power comparison between the ECF test and the EDF tests is conducted.

The organization of this article is as follows. Section 2 discusses estimation of the unknown parameters of the GE(



, ) by the ML method. The goodness-fit-test based on ECF for the GE(



, ) is provided in section 3. EDF tests for the GE(



, ) are discussed in Section 4. Then, a power study is conducted at section 5. Finally, conclusions are included in section 6.

2. ML Estimation of the unknown parameters of the Generalized Exponential distribution

For testing H0 given in (1.4) the unknown parameters of the GE(



,) must be

estimated first. In this section the ML method for parameter estimation is considered.

Gupta and Kundu (2001b) estimated the two parameters,



and



, of the GE(



, ) using the ML method. They concluded the following two equations:

ˆ

ˆ

ln(1 ) n

x

n e 



  



ˆ ˆ ˆ ˆ

1 1 1 1

ˆ{[ ln(1 i)][ {( i) (1 i)}] [ ][ ln(1 i)]}

n n n n

x x x x

i i

i i i i

n e  x e  e  x e 

    

   









 

 





ˆ



1ln 1 0

i n

x

i

n e







  . (2.2)

Where x x₁, ,...,₂ xn is a random sample from the GE(



, ). Equation (2.2) has to

be solved for the unknown



ˆ by iteration to obtain the ML estimator of . Then, the ML estimator of



, ˆ, can be easily obtained as a function of



ˆ from equation (2.1).

3. Goodness-of-fit tests based on ECF for the generalized exponential distribution

There are different forms of test statistics that are based on ECF. These tests depend on measuring the distance between the ECF of the sample data and the CF of the hypothesized distribution. Therefore; according to the technique of measuring this distance; these tests can be divided into two major types. First; tests based on the integral of the squared modulus of the difference between the ECF and the CF of the null distribution, such as the test of Wong and Sim (2000). Second, tests based on the difference between the real (or imaginary) part of the ECF and the real (or imaginary) part of the CF of the hypothesized distribution, such as the test of Towhidi and Salmanpour (2007).

The first type of these tests can be applied only for distributions which have CF in closed form but for other distributions which do not have CF in closed form the computations will be difficult if not impossible. This is because the algorithm will contain more than one integration that does not converge to a solution. On the other hand, the second type of these tests can be applied for any distribution whether its CF is in closed form or not.

Since the CF of the generalized exponential distribution is not in a closed form, then the test of Towhidi and Salmanpour (2007) is the most appropriate in this case. This test is explained in details as follows.

The test of Towhidi and Salmanpour (2007) is given by:









* 1

0 0

( ) ( ) ( ) ( )

n n m m k k k n m m

T _ Z T _Z T B _B Z T_{ }Z T _(2.3)

Where

1

2

1

2

Re ( ) Re ( )

Re ( ) ( )

Im ( ) Im ( )

Im ( ) n n n m n m n n t t t Z T t t t     _       _      _                  , 1 2 0 1 2

Re ( ) Re ( )

Re ( ) ( )

Im ( ) Im ( )

i) Tm ( , ,..., )t t1 2 tm is an arbitrary point in m, where m is a positive integer

that is equal to the sample sizen.

ii) ( )t is the CF of the hypothesized distribution and n( )t is the ECF.

iii) Imn( )t =

1

1 n _{sin( )} j j

tX n





and Ren( )t =

1

1 n _{cos( )} j j

tX n





are the imaginary

and real parts of n( )t , respectively.

iv) Im ( )t =E[sin( )]tX and Re ( ) t =E[cos( )]tX are the imaginary and real parts of ( )t , respectively.

v) k is a k k diagonal matrix whose diagonal elements,  1, ,...,2 k , are

the k largest eigen values of 0.0 be the 2m2m variance-covariance

matrix

of Y Tj( )m under H0. Where

1 1

1 1

cos( ) cos( )

cos( ) cos( ) ( )

sin( ) sin( )

sin( ) sin( )

j j

m j m j

j m

j j

m j m j

t X E t X

t X E t X

Y T

t X E t X

t X E t X

 _{ } _       _{ }            _{ }       _{  }      (2.5)

And the integer value k is chosen such that the ratio 2

1 1 k m j j j j





 





is near

to 1.

vi) Bk is a 2m k matrix whose columns are the eigen vectors, B B1, 2,...,Bk,

corresponding to  ₁, ,...,₂ k.

It is known that the standard procedure for testing H0 , given in (1.4), is to

compare the value of the test statistic for the observed sample with its theoretical critical values (percentage points) under H0.

The exact distribution of the test statistic, *

n

T , under H0 is difficult; if not

impossible; to obtain. In this section, Monte Carlo procedures are employed to obtain the empirical critical values of *

n

T under the GE( , ) null distribution. Two significance levels will be considered that are  0.05 and 0.01. Different sample sizes are considered n 10, 20 and 30 with N= 10,000 replications. In addition, different values for the shape parameter



were considered,

0.5,1,1.5, 2, 2.5,

To find the empirical critical values for the different significance levels, the following steps are performed:

Step (1): Generate N random samples of size n from the GE( , ). This can be done by firstly generating a random sample from the uniform(0,1) distribution,

n

u u

u1, 2,..., . Then the uniform random numbers can be transformed to

generalized exponential random numbers using the following transformation:

 

] 1

ln[ ) 1

( 1

 i

i u

x    , i 1,2,…,n (2.6)

Step (2): Estimate the parameters of the GE( , ) using the ML method. Step (3): Calculate the value of *

n

T for each random sample. This is done as follows:

i) Select the values t₁,...,tm by the relation: ti F1_mi_1, i 1,...,m

  ,

where _F1 _{is the inverse distribution function of the GE( , ) and}

m n (the sample size).

ii) Define the vector Tm 



t1,...,tm



and evaluate the vector Z Tn

 

m Z T0

 

m .

Where Z Tn

 

m and Z T0

 

m are defined in (2.4).

iii) Determine the matrix 0 which is the 2m2m variance-covariance matrix

of Y Tj( )m , which is defined in (2.5), under H0. Then derive its eigen

values and eigen vectors.

iv) Arrange the eigen values, i, such that 12  ... 2m. Then, choose an

integer k so that the ratio 1 2

1

k i i

m i i





 





is large enough (near to one).

Construct the matrices k and Bk.

v) Calculate the value of *

n

T as in (2.3).

Step (4): Arrange the Nvalues of *

n

T in an ascending order Step (5): Calculate the 100(1- )thpercentile from the ordered *

n

T values.

4. Goodness-of-fit tests based on EDF for the generalized exponential distribution

The most commonly used goodness-of-tests are the tests based on EDF. These test statistics generally measure in a different way the distance between a CDF;

0( )

F x ; and the EDF ; denoted byF xn( ); which is calculated from the sample.

There are many tests based on EDF statistics such as the K-S, C-M and A-D tests. In the last two decades, many authors (Lawless, 1982; Liao and Shimokawa, 1999b; Littell et al., 1979; Park and Seoh, 1994; Stephens, 1974) have reported that the A-D and C-M tests are more powerful than the K-S test. At present, the A-D test is mostly applied in the statistical analysis of lifetime data (Liao and Shimokawa, 1999a).

When F x0( ) is completely specified (i.e. does not contain unknown parameters)

and the data are uncensored, the EDF tests are all distribution free and percentage points for these test statistics are generally known. However, this is no longer the case when data are censored or F x₀( ) involves unknown parameters. Therefore; if the hypothesized distribution contains unknown parameters that must be estimated from the sample data, then the standard tables used for the EDF statistics are no longer valid and the resulting test would be inaccurate. In this case, these statistics can be modified by inserting estimates of unknown parameters in F x₀( ). However, it is not possible to obtain an analytic expression for the distribution of these modified statistics. Hence, the critical values for these modified statistics are usually obtained by Monte Carlo simulation methods (Abd-Elfattah et al., 2010).

In this section; the empirical percentage points of several modified EDF statistics, under the null hypothesis that the data are from the GE(



,), will be obtained by Monte Carlo simulations. The considered EDF statistics are:

1. The K-S test statistic:





max ,

D _ D D  _{, (4.1)}

where:D maxi



 

i n F x0



_{ }i , , ˆ ˆ





 _{ }

and

 









0 ˆ ˆ 1

maxi i , , i

D _ F x

 

_  _n 

 _{ }

 

 





0 i, ,ˆ ˆ

2. The C-M statistic:

 





2

2

0 1

1 _{, ,ˆ ˆ} 2 1

12 2

n

i i

i

W F x

n    n



 

  _  _

 



. (4.2)

3. The WA statistic:

 





2

0

2 2 1

ˆ ˆ

, , ₁ 2 n

i i F x

U W n                  



. (4.3)

4. The A-D statistic:









_{ }





_{ }





2

0 0 1

1

1 n _{2 1 ln , ,ˆ ˆ ln 1 , ,ˆ ˆ}

i n i

i

A n i F x F x

n      



 

  



  _  _ . (4.4)

5. The L-S statistic:

 







 



 







 



0 0 1 0 0 1 ˆ ˆ ˆ ˆ

max , , , , ,

1

ˆ ˆ

ˆ ˆ

, , 1 , ,

i i

n n

i

i i

i _{F x F x} i

n n

L

n _{F x F x}

 

 

 

 

   _{ }        _   



. (4.5)

The steps for obtaining critical values for these five EDF statistics are the same steps followed for obtaining critical values for the *

n

T statistic in the previous section. Except for step (5) where the D,_W 2_{, U}2_{, A}2 _and

n

L statistics must be calculated from each sample. These steps were performed using the Mathcad (version 13) software. Different sample sizes were considered: n 10, 20 and 30. Two significance levels considered:  0.05 and 0.01. For the GE(



, ); different values for the shape parameter



were considered that are:



_{0.5(0.5) 3, while}  was always considered to be 1.

The critical values for the different modified EDF statistics are listed in table (1) in the appendix with the critical values of the *

n

Table (1): Critical values of the EDF and ECF Statistics

n Statistic  Value of_0.5



₁ the shape parameter of the null distribution_{1.5 2 2.5 3}

10

D 0.05 0.748 0.847 0.899 0.938 0.962 0.976

0.01 0.942 0.991 0.999 1 1 1

2

W 0.05_0.01 2.079 2.679 2.97_{3.071 3.303 3.33} 3.126_3.333 3.213_3.333 3.262_3.333

2

A 0.05 13.303 22.269 31.263 40.109_{0.01 30.878 59.165 90.529 125.136 164.164}49.083 _207.31658.086

2

U 0.05_0.01 1.917 2.472 2.743_{2.839 3.054 3.08} 2.889_{3.083 3.083}2.97 3.016_3.083

n

L 0.05 10.64 18.60 31.64 57.34 104.83 202.07

0.01 103.44 253.50 1344.43 13997.04 226276.89 5730672.82

*

n

T 0.05 9.044 8.957 8.38 8.38 8.267 8.114

0.01 50.361 22.702 20.467 20.467 21.183 19.106

20

D 0.05_0.01 0.489 0.569 0.621_{0.660 0.759 0.818} 0.660_0.857 0.690_0.884 0.714_0.906

2

W 0.05_0.01 1.827 2.627 3.187_{3.432 4.636 5.280} 3.616_5.675 3.957_5.947 4.238_6.131

2

A 0.05 10.012 15.377 20.063 24.373_{0.01 19.671 32.797 44.691 55.855} 28.544_67.393 32.466_78.512

2

U 0.05_0.01 1.753 2.521 3.057_{3.297 4.455 5.075} 3.469_5.455 3.798_5.718 4.068_5.897

n

L 0.05 4.964 6.372 7.917 9.440 11.199 13.068

0.01 16.043 17.001 22.258 30.864 45.057 66.663

*

n

T 0.05 2.942 3.598 4.582 4.831 4.968 4.61

0.01 9.754 11.161 10.644 10.296 9.437 8.658

30

D 0.05 0.388 0.453 0.497 0.529 0.554 0.570

0.01 0.512 0.599 0.652 0.688 0.718 0.740

2

W 0.05 1.668 2.436 3.022 3.498 3.892 4.155

0.01 3.113 4.509 5.464 6.131 6.611 6.846

2

A 0.05_{0.01 16.647 26.291 34.883 43.131}9.178 13.797 17.530 21.200 24.235_50.966 26.484_55.744

2

U 0.05 1.625 2.372 2.941 3.407 3.788 4.040

0.01 3.031 4.388 5.319 5.969 6.437 6.665

n

L 0.05 3.971 4.943 5.786 6.650 7.420 7.844

0.01 8.085 9.499 11.323 13.282 15.796 17.586

*

n

T 0.05 1.687 1.833 2.298 2.552 2.669 2.742

5. Power Study

The power of a goodness-of-fit test is defined as the probability that a statistic will lead to the rejection of the null hypothesis, H0, when it is false. The power of a

goodness-of-fit test at a significance level  is denoted by 1, where  is the probability of committing a type-II error, that is failing to reject a false null hypothesis.

In this section a power comparison is conducted between the different EDF tests and the ECF test of Towhidi and Salmanpour (2007), for the GE( , ) with unknown parameters. For the alternative hypothesis, Ha, that the random

sample comes from another distribution; several alternative distributions were considered. These distributions are:

i) Chi-Square distribution with shape parameter



_{(=1.5 and 3).}

ii) Exponential distribution with scale parameter (=1).

iii) Gamma distribution with shape parameter



(=1.5 and 2.5). iv) Lognormal distribution with scale parameter (=1).

v) Weibull distribution with shape parameter



(=1.5).

The steps for obtaining the power of any test are as follows:

Step (1): Generate N (=10,000) random samples of size n from each alternative distribution.

Step (2): Calculate the value of the test statistic from each sample. Step (3): Arrange theNvalues of the test statistic in an ascending order.

Step (4): From table (1), and for each significance level ( ); get the critical value, call itc, of each test statistic under the null distribution GE(



,).

Step (5): Calculate L = the number of the test statistic values that are greater thanc.

Step (6): Calculate the power of the test:

1= L_N

These steps were performed using the Mathcad (version 13) software. Different sample sizes were considered: n10, 20 and 30 with two significance levels:

 0.01 and 0.05. The results of the power comparison are shown, for each alternative distribution, in tables (2) to (9).

Table (2): Power values when the alternative distribution is Chi-square (



_1.5)

n Test  Value ofof the null distribution



the shape parameter

0.5 1 1.5 2 2.5 3

10

D 0.05 0.277 0.007 0.004 0.003 0.002 0.002

0.01 0.003 0.001 0.001 0 0 0

2

W 0.05_0.01 0.022_0.003 0.006_0.001 0.004_0.001 0.003₀ 0.002₀ 0.001₀

2

A 0.05_0.01 0.205_0.049 0.090_0.024 0.048_0.020 0.032_0.005 0.026_0.003 0.024_0.003

2

U 0.05_0.01 0.022_0.003 0.006_0.001 0.004_0.001 0.003₀ 0.002₀ 0.001₀

n

L 0.05 0.366 0.302 0.256 0.217 0.185 0.158

0.01 0.185 0.149 0.101 0.062 0.037 0.022

*

n

T 0.05 0.247 0.247 0.256 0.256 0.257 0.259

0.01 0.105 0.161 0.169 0.169 0.166 0.175

20

D 0.05 0.264 0.107 0.045 0.019 0.008 0.004

0.01 0.019 0.001 0 0 0 0

2

W 0.05_0.01 0.248_{0.011 0.0002}0.075 0.022₀ 0.007₀ 0.003₀ 0.001₀

2

A 0.05_0.01 0.484_0.238 0.328_0.092 0.233_0.044 0.167_0.027 0.125_0.017 0.094_0.011

2

U 0.05_0.01 0.248_0.011 0.075₀ 0.022₀ 0.007₀ 0.003₀ 0.001₀

n

L 0.05 0.600 0.543 0.496 0.460 0.428 0.400

0.01 0.368 0.359 0.325 0.292 0.260 0.229

*

n

T 0.05 0.579 0.543 0.494 0.483 0.479 0.493

0.01 0.340 0.311 0.322 0.328 0.346 0.366

30

D 0.05_0.01 0.503_0.146 0.299_0.029 0.182_0.005 0.115_0.002 0.076_{0.001 0.0002}0.056

2

W 0.05_0.01 0.527_0.148 0.297_0.014 0.165_{0.001 0.0001}0.089 0.046₀ 0.029₀

2

A 0.05_0.01 0.678_0.457 0.537_0.255 0.434_0.144 0.348_0.085 0.288_0.050 0.251_0.037

2

U 0.05_0.01 0.527_0.149 0.297_0.014 0.166_{0.001 0.0001}0.089 0.047₀ 0.030₀

n

L 0.05 0.751 0.694 0.645 0.606 0.579 0.562

0.01 0.554 0.518 0.476 0.441 0.410 0.391

*

n

T 0.05 0.790 0.777 0.745 0.727 0.719 0.714

Table (3): Power values when the alternative distribution is Chi-square (



₃₎

n Test  Value ofof the null distribution



the shape parameter

0.5 1 1.5 2 2.5 3

10

D 0.05_0.01 0.332_0.009 0.104_0.001 0.023₀ 0.010₀ 0.005₀ 0.003₀

2

W 0.05_0.01 0.337_0.017 0.078_0.001 0.028₀ 0.012₀ 0.006₀ 0.003₀

2

A 0.05_0.01 0.657_0.336 0.476_0.119 0.331_0.082 0.229_0.058 0.162_0.024 0.123_0.020

2

U 0.05 0.337 0.078 0.028 0.012 0.006 0.003

0.01 0.017 0.001 0 0 0 0

n

L 0.05 0.716 0.643 0.579 0.517 0.461 0.410

0.01 0.462 0.394 0.296 0.198 0.125 0.080

*

n

T 0.05 0.586 0.588 0.599 0.599 0.600 0.604

0.01 0.241 0.397 0.418 0.418 0.412 0.433

20

D 0.05_0.01 0.906_0.570 0.800_0.225 0.689_0.075 0.570_0.025 0.473_0.007 0.385_0.002

2

W 0.05_0.01 0.918_0.610 0.811_0.185 0.687_0.041 0.551_0.008 0.430_0.002 0.324_0.001

2

A 0.05_0.01 0.952_0.864 0.909_0.699 0.860_0.526 0.811_0.394 0.755_0.282 0.704_0.206

2

U 0.05_0.01 0.918_0.609 0.811_0.185 0.688_0.041 0.552_0.008 0.431_0.002 0.325_0.001

n

L 0.05 0.954 0.935 0.915 0.899 0.878 0.857

0.01 0.830 0.823 0.786 0.744 0.690 0.639

*

n

T 0.05 0.928 0.913 0.892 0.886 0.884 0.891

0.01 0.795 0.767 0.777 0.783 0.799 0.814

30

D 0.05 0.984 0.964 0.936 0.908 0.873 0.847

0.01 0.924 0.783 0.617 0.471 0.339 0.246

2

W 0.05_0.01 0.988_0.943 0.972_0.797 0.947_0.595 0.918_0.395 0.882_0.250 0.851_0.182

2

A 0.05_0.01 0.991_0.975 0.982_0.936 0.973_0.884 0.960_0.818 0.947_0.743 0.935_0.691

2

U 0.05_0.01 0.988_0.943 0.972_0.798 0.947_0.595 0.918_0.395 0.881_0.251 0.851_0.183

n

L 0.05 0.992 0.987 0.982 0.977 0.973 0.969

0.01 0.968 0.956 0.943 0.928 0.911 0.899

*

n

T 0.05 0.948 0.945 0.931 0.925 0.920 0.919

Table (4): Power values when the alternative distribution is Exponential (₁₎

n Test  Value ofof the null distribution



the shape parameter

0.5 1 1.5 2 2.5 3

10

D 0.05_0.01 0.087_0.025 0.050_0.010 0.035_0.005 0.026_0.000 0.019_0.000 0.015_0.000

2

W 0.05_0.01 0.092_0.027 0.050_0.010 0.034_0.005 0.024_0.003 0.018_0.003 0.013_0.003

2

A 0.05_0.01 0.101_0.031 0.050_0.010 0.030_0.005 0.020_0.002 0.014_0.001 0.010_0.001

2

U 0.05_0.01 0.092_0.027 0.050_0.010 0.034_0.005 0.024_0.003 0.018_0.003 0.013_0.003

n

L 0.05 0.082 0.050 0.034 0.023 0.016 0.012

0.01 0.016 0.011 0.005 0.002 0.001 0.001

*

n

T 0.05_0.01 0.463_0.186 0.465_0.305 0.476_0.323 0.476_0.323 0.479_0.317 0.483_0.332

20

D 0.05_0.01 0.090_0.025 0.052_0.010 0.034_0.005 0.025_0.003 0.019_0.002 0.015_0.001

2

W 0.05_0.01 0.091_0.027 0.050_0.009 0.032_0.005 0.023_0.003 0.017_0.002 0.013_0.001

2

A 0.05_0.01 0.101_0.031 0.051_0.009 0.029_0.004 0.018_0.002 0.012_0.001 0.009_0.001

2

U 0.05_0.01 0.091_0.027 0.050_0.009 0.032_0.005 0.023_0.003 0.017_0.002 0.013_0.001

n

L 0.05 0.084 0.050 0.030 0.022 0.017 0.012

0.01 0.010 0.009 0.006 0.004 0.003 0.002

*

n

T 0.05 0.099 0.079 0.060 0.056 0.055 0.059

0.01 0.021 0.017 0.018 0.019 0.022 0.025

30

D 0.05 0.093 0.048 0.032 0.024 0.018 0.016

0.01 0.027 0.011 0.007 0.004 0.003 0.002

2

W 0.05_0.01 0.095_0.032 0.050_0.011 0.034_0.006 0.024_0.003 0.019_0.002 0.015_0.002

2

A 0.05_0.01 0.100_0.035 0.049_0.011 0.030_0.005 0.020_0.002 0.015_0.001 0.011_0.001

2

U 0.05_0.01 0.094_0.032 0.050_0.011 0.034_0.006 0.024_0.003 0.019_0.002 0.015_0.002

n

L 0.05 0.093 0.052 0.034 0.023 0.016 0.014

0.01 0.013 0.009 0.006 0.004 0.003 0.002

*

Table (5): Power values when the alternative distribution is Exponential (2)

n Test  Value ofof the null distribution



the shape parameter

0.5 1 1.5 2 2.5 3

10

D 0.05_0.01 0.380_0.130 0.248_0.054 0.184_0.024 0.135_0.000 0.103_0.000 0.082_0.000

2

W 0.05_0.01 0.388_0.136 0.240_0.051 0.164_0.024 0.120_0.014 0.090_0.014 0.072_0.014

2

A 0.05_0.01 0.380_0.138 0.218_0.045 0.136_0.019 0.090_0.009 0.063_0.005 0.047_0.004

2

U 0.05_0.01 0.388_0.136 0.240_0.052 0.165_0.025 0.120_0.014 0.091_0.014 0.072_0.014

n

L 0.05 0.213 0.128 0.083 0.056 0.039 0.028

0.01 0.040 0.025 0.013 0.006 0.004 0.002

*

n

T 0.05_0.01 0.798_0.480 0.799_0.668 0.807_0.687 0.807_0.687 0.808_0.679 0.810_0.697

20

D 0.05_0.01 0.669_0.360 0.526_0.203 0.434_0.128 0.360_0.085 0.309_0.063 0.267_0.047

2

W 0.05_0.01 0.680_0.374 0.523_0.189 0.419_0.112 0.342_0.074 0.287_0.050 0.246_0.036

2

A 0.05_0.01 0.659_0.374 0.485_0.168 0.365_0.088 0.281_0.048 0.218_0.028 0.171_0.017

2

U 0.05_0.01 0.680_0.373 0.523_0.189 0.420_0.112 0.342_0.074 0.288_0.051 0.246_0.036

n

L 0.05 0.478 0.329 0.230 0.165 0.123 0.094

0.01 0.064 0.058 0.036 0.022 0.012 0.007

*

n

T 0.05 0.240 0.197 0.156 0.147 0.144 0.155

0.01 0.064 0.054 0.058 0.060 0.068 0.076

30

D 0.05_0.01 0.839_0.597 0.723_0.405 0.630_0.293 0.561_0.225 0.505_0.177 0.469_0.145

2

W 0.05_0.01 0.853_0.622 0.739_0.403 0.638_0.269 0.559_0.193 0.497_0.148 0.456_0.128

2

A 0.05_0.01 0.831_0.606 0.693_0.370 0.582_0.228 0.485_0.141 0.415_0.088 0.366_0.066

2

U 0.05_0.01 0.852_0.621 0.738_0.403 0.638_0.269 0.558_0.193 0.497_0.148 0.456_0.129

n

L 0.05_0.01 0.723_0.225 0.563_0.152 0.444_0.095 0.343_0.061 0.273_0.036 0.242_0.026

*

n

Table (6): Power values when the alternative distribution is Gamma (



_1.5)

n Test  Value ofof the null distribution



the shape parameter

0.5 1 1.5 2 2.5 3

10

D 0.05_0.01 0.075_0.022 0.045_0.011 0.032_0.006 0.024_0.000 0.019_0.000 0.016_0.000

2

W 0.05_0.01 0.077_0.026 0.045_0.012 0.031_0.006 0.023_0.004 0.018_0.004 0.015_0.004

2

A 0.05_0.01 0.111_0.034 0.055_0.013 0.033_0.007 0.023_0.004 0.017_0.003 0.013_0.002

2

U 0.05_0.01 0.077_0.025 0.045_0.012 0.031_0.006 0.023_0.004 0.018_0.004 0.015_0.004

n

L 0.05_0.01 0.121_0.028 0.079_0.019 0.054_0.010 0.038_0.005 0.028_0.003 0.021_0.002

*

n

T 0.05_0.01 0.301_0.050 0.303_0.149 0.316_0.166 0.316_0.166 0.319_0.161 0.323_0.174

20

D 0.05_0.01 0.084_0.016 0.037_0.007 0.022_0.004 0.016_0.003 0.011_0.002 0.009_0.002

2

W 0.05_0.01 0.082_0.017 0.034_0.007 0.021_0.004 0.014_0.003 0.011_0.002 0.009_0.002

2

A 0.05_0.01 0.131_0.034 0.063_0.010 0.033_0.004 0.020_0.003 0.014_0.002 0.011_0.002

2

U 0.05_0.01 0.082_0.017 0.034_0.007 0.021_0.004 0.014_0.003 0.011_0.002 0.009_0.002

n

L 0.05_0.01 0.146_0.027 0.099_0.025 0.068_0.020 0.052_0.014 0.041_0.011 0.034_0.007

*

n

T 0.05_0.01 0.279_0.102 0.244_0.086 0.204_0.091 0.196_0.095 0.193_0.105 0.204_0.115

30

D 0.05_0.01 0.115_0.020 0.048_0.006 0.025_0.003 0.016_0.002 0.011_0.001 0.008_0.001

2

W 0.05_0.01 0.121_0.022 0.045_0.006 0.024_0.003 0.014_0.002 0.010_0.001 0.008_0.001

2

A 0.05_0.01 0.169_0.054 0.081_0.016 0.047_0.006 0.028_0.002 0.019_0.002 0.015_0.001

2

U 0.05_0.01 0.121_0.022 0.045_0.006 0.024_0.003 0.014_0.002 0.010_0.001 0.008_0.001

n

L 0.05 0.183 0.120 0.085 0.064 0.052 0.047

0.01 0.044 0.033 0.024 0.018 0.014 0.012

*

n

Table (7): Power values when the alternative distribution is Gamma (



_2.5)

n Test  Value ofof the null distribution



the shape parameter

0.5 1 1.5 2 2.5 3

10

D 0.05 0.090 0.033 0.020 0.016 0.013 0.012

0.01 0.016 0.009 0.006 0.000 0.000 0.000

2

W 0.05_0.01 0.086_0.018 0.030_0.009 0.021_0.006 0.016_0.004 0.014_0.004 0.012_0.004

2

A 0.05_0.01 0.236_0.053 0.106_0.014 0.051_0.008 0.030_0.006 0.020_0.004 0.015_0.003

2

U 0.05_0.01 0.086_0.018 0.030_0.009 0.021_0.006 0.016_0.004 0.014_0.004 0.012_0.004

n

L 0.05 0.262 0.185 0.135 0.098 0.072 0.053

0.01 0.072 0.048 0.025 0.012 0.006 0.004

*

n

T 0.05 0.220 0.221 0.230 0.230 0.232 0.235

0.01 0.069 0.121 0.129 0.129 0.125 0.134

20

D 0.05 0.390 0.218 0.125 0.073 0.042 0.027

0.01 0.073 0.009 0.003 0.001 0.001 0.001

2

W 0.05_0.01 0.407_0.077 0.215_0.005 0.110_0.002 0.057_0.001 0.029_0.001 0.014_0.001

2

A 0.05_0.01 0.494_0.255 0.348_0.087 0.248_0.031 0.179_0.013 0.127_0.005 0.090_0.002

2

U 0.05_0.01 0.407_0.076 0.215_0.005 0.111_0.002 0.057_0.001 0.029_0.001 0.014_0.001

n

L 0.05 0.492 0.404 0.328 0.277 0.237 0.205

0.01 0.168 0.158 0.120 0.089 0.064 0.046

*

n

T 0.05 0.253 0.222 0.184 0.177 0.173 0.183

0.01 0.093 0.080 0.085 0.088 0.096 0.104

30

D 0.05 0.601 0.442 0.322 0.246 0.192 0.162

0.01 0.285 0.107 0.041 0.017 0.006 0.003

2

W 0.05_0.01 0.630_0.321 0.462_0.106 0.340_0.028 0.249_0.006 0.185_0.001 0.148_0.001

2

A 0.05_0.01 0.671_0.470 0.543_0.268 0.448_0.158 0.364_0.090 0.303_0.047 0.264_0.032

2

U 0.05_0.01 0.630_0.320 0.462_0.107 0.340_0.028 0.248_0.006 0.185_0.001 0.148_0.001

n

L 0.05 0.668 0.580 0.506 0.444 0.395 0.372

0.01 0.360 0.301 0.242 0.200 0.164 0.147

*

n

T 0.05 0.567 0.554 0.517 0.506 0.501 0.499

Table (8): Power values when the alternative distribution is Lognormal (1)

n Test  Value ofof the null distribution



the shape parameter

0.5 1 1.5 2 2.5 3

10

D 0.05_0.01 0.114_0.027 0.059_0.012 0.038_0.006 0.028_0.000 0.022_0.000 0.017_0.000

2

W 0.05_0.01 0.127_0.032 0.062_0.012 0.040_0.006 0.028_0.004 0.021_0.004 0.017_0.004

2

A 0.05_0.01 0.230_0.101 0.138_0.070 0.100_0.057 0.084_0.018 0.075_0.013 0.070_0.012

2

U 0.05_0.01 0.126_0.032 0.062_0.012 0.039_0.006 0.028_0.004 0.021_0.004 0.017_0.004

n

L 0.05 0.326 0.274 0.240 0.211 0.188 0.167

0.01 0.189 0.161 0.128 0.098 0.074 0.058

*

n

T 0.05 0.602 0.602 0.609 0.609 0.610 0.611

0.01 0.500 0.542 0.548 0.548 0.546 0.552

20

D 0.05 0.222 0.126 0.082 0.057 0.041 0.033

0.01 0.057 0.020 0.009 0.006 0.004 0.003

2

W 0.05_0.01 0.246_0.071 0.140_0.021 0.088_0.011 0.059_0.007 0.043_0.004 0.033_0.003

2

A 0.05_0.01 0.341_0.180 0.236_0.101 0.176_0.078 0.140_0.068 0.119_0.049 0.103_0.033

2

U 0.05_0.01 0.245_0.071 0.140_0.021 0.088_0.010 0.059_0.007 0.043_0.004 0.033_0.003

n

L 0.05 0.450 0.405 0.371 0.348 0.329 0.315

0.01 0.296 0.292 0.275 0.252 0.234 0.219

*

n

T 0.05_0.01 0.403_0.152 0.357_0.128 0.305_0.136 0.292_0.141 0.287_0.158 0.303_0.175

30

D 0.05_0.01 0.301_0.117 0.194_0.048 0.134_0.025 0.100_0.013 0.080_0.008 0.068_0.006

2

W 0.05_0.01 0.346_0.152 0.227_0.063 0.162_0.029 0.121_0.014 0.091_0.009 0.077_0.007

2

A 0.05_0.01 0.427_0.268 0.319_0.162 0.255_0.113 0.210_0.088 0.178_0.074 0.160_0.065

2

U 0.05_0.01 0.345_0.152 0.226_0.063 0.162_0.029 0.120_0.013 0.091_0.009 0.077_0.007

n

L 0.05 0.537 0.489 0.458 0.432 0.415 0.407

0.01 0.402 0.379 0.360 0.346 0.332 0.323

*

Table (9): Power values when the alternative distribution is Weibull (



_1.5)

n Test  Value ofof the null distribution



the shape parameter

0.5 1 1.5 2 2.5 3

10

D 0.05 0.442 0.3198 0.253 0.203 0.168 0.142

0.01 0.198 0.103 0.058 0.000 0.000 0.000

2

W 0.05_0.01 0.457 0.3204 0.244_{0.212 0.104 0.060 0.0402 0.0402 0.0402}0.194 0.159 0.134

2

A 0.05_0.01 0.466_0.233 0.318_0.105 0.229_0.056 0.173_0.035 0.135_0.023 0.108_0.016

2

U 0.05_0.01 0.457 0.3204 0.244_{0.212 0.104 0.061 0.0405 0.0405 0.0405}0.194 0.159 0.135

n

L 0.05 0.335 0.240 0.180 0.137 0.107 0.083

0.01 0.108 0.077 0.049 0.030 0.019 0.013

*

n

T 0.05 0.061 0.063 0.083 0.083 0.086 0.092

0.01 0.001 0.001 0.001 0.001 0.001 0.001

20

D 0.05 0.696 0.568 0.481 0.418 0.370 0.333

0.01 0.418 0.267 0.185 0.137 0.108 0.085

2

W 0.05_0.01 0.714_0.442 0.573_0.264 0.482_0.179 0.413_0.131 0.361_0.100 0.317_0.079

2

A 0.05_0.01 0.707_0.461 0.556_0.262 0.453_0.165 0.374_0.111 0.311_0.076 0.265_0.053

2

U 0.05_0.01 0.713_0.442 0.573_0.264 0.482_0.180 0.413_0.131 0.361_0.100 0.318_0.079

n

L 0.05 0.577 0.454 0.352 0.285 0.234 0.193

0.01 0.152 0.141 0.105 0.074 0.048 0.033

*

n

T 0.05 0.466 0.409 0.343 0.330 0.324 0.342

0.01 0.150 0.118 0.129 0.136 0.158 0.180

30

D 0.05 0.828 0.733 0.655 0.591 0.542 0.509

0.01 0.624 0.448 0.343 0.276 0.227 0.193

2

W 0.05_0.01 0.846_0.654 0.749_0.458 0.667_0.336 0.604_0.257 0.549_0.210 0.509_0.188

2

A 0.05_0.01 0.834_0.654 0.722_0.451 0.636_0.316 0.556_0.223 0.491_0.164 0.447_0.135

2

U 0.05_0.01 0.845_0.653 0.748_0.458 0.667_0.336 0.604_0.257 0.549_0.210 0.509_0.188

n

L 0.05 0.769 0.649 0.553 0.460 0.392 0.361

0.01 0.345 0.263 0.196 0.145 0.106 0.086

*

n

When comparing the power values of the EDF tests with the ECF test, from tables (2) to (9), it can be noticed that:

1. When n 10, The ECF test is the most powerful test in most of the cases. Some exceptions happen as in the case for the Weibull ( 1.5) alternative where it can be seen that the EDF tests are powerful than the ECF test.

2. When n 20,

i. The ECF test is the most powerful test when the alternative distribution is gamma ( 1.5). Also, it is the most powerful test when the alternative is the exponential (1) except when the shape parameter of the null distribution  0.5 where the A-D test has the highest power value.

ii. The L-S test is the most powerful test when the alternative distribution is lognormal. Also, it is the most powerful test when the alternative is the Chi-square ( 3) distribution except when  =0.01 and the shape parameter of the null distribution  0.5 where the A-D test has the highest power value.

iii. For the Chi-square ( 1.5) alternative, the ECF test is the most powerful when the shape parameter of the null distribution  2, 2.5 and 3. On the other hand, the L-S test is the most powerful test when the shape parameter of the null distribution  0.5, 1 and 1.5.

iv. For the Chi-square ( 3) alternative, exponential (2) and Weibull ( 1.5) alternatives, the EDF tests have higher power than the ECF test. Except for large values of the shape parameter value of the null distribution



_{2.5 and 3 where the ECF test has higher}

power.

3. When n 30,

i. The ECF test is the most powerful test when the alternatives are the Chi-square ( 1.5) and exponential





1



_.

ii. The L-S test is the most powerful test when the alternative is the lognormal distribution.

iii. For the Chi-square ( 3) alternative; the L-S test is the most powerful except when  0.01 and the shape parameter of the null distribution is  0.5 where the A-D test is the most powerful.

iv. For the exponential ( 2) alternative, the following is noticed:

a. At  0.05 and when the shape parameter of the null distribution is  0.5,1 and 1.5; the C-M test is the most powerful. In addition; when the shape parameter of the null distribution is

b. At  0.01 and when the shape parameter of the null distribution is  0.5; the C-M test is the most powerful. In addition; when the shape parameter of the null distribution is



1 the K-S test is the most powerful.

c. For the rest of the cases, the ECF test has the greatest power. v. For the gamma





1.5



alternative; the ECF test is the most

powerful test in most of the cases except when the value of the shape parameter of the null distribution is



0.5 where the L-S test is the most powerful at  0.05, but, when  0.01 the A-D test is the most powerful.

vi. For the gamma





2.5



alternative; the ECF test is the most powerful test in most of the cases except when the value of the shape parameter of the null distribution is



0.5 where the A-D test is the most powerful. Also, when  0.05 and



1 the L-S test is the most powerful.

vii. For the Weibull ( 1.5) alternative, it can be seen that the EDF tests are powerful than the ECF test. The only exception happens is when the value of the shape parameter of the null distribution is  3, where the ECF test is the most powerful.

6. Conclusions

Generally it can be concluded that there is no one test that is optimal for all of the cases. It was shown that the most powerful test depends on the sample size, the alternative distribution, the significance level and the value of the shape parameter of the null distribution. Also, it is concluded that the ECF test is the best when the sample size is small (n 10) except when the alternative distribution is Weibull (



1.5) where the EDF tests are better.

References

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